Question

A uniform sphere of mass m and radius r is set free from the top edge of a semicircle halfpipe with radius R. If R > r, what is the time-dependent velocity equation v(t) for the sphere in terms of t, m, r, R and g ignoring any effects of friction?

Could anybody help me derive the differential equations?

Answer 1

i think you can use conservation of energy principle!!

Answer 2

I doubt \[mgh = \frac{ 1 }{ 2 }mv^2\] will get me something in terms of t

Answer 3

No, this won't get you somewhere, because the potential gravitational energy will _not_ be "transformed" into kinetic energy to 100%. There will be rotational energy, too.. you have to take that into account..

Answer 4

yea.. total kinetic energy = rotational kinectic + translational kinetic.. thats not all that hard cause moment of inertia of the sphere would be known.. and m assuming its without slip!!

Answer 5

If it were an inclined plane, we could stick with \[gtsin \alpha\] with alpha being the angle of inclination.
If we consider a semicircle as an infinitesimal sum of inclined planes, we could do
\[\int\limits_{0}^{t}gsin \alpha(t) d \alpha\]
But it's quite a challenge to derive alpha(t). Not even sure if possible

Answer 6

Shouldn't the acceleration in terms of the angle look like this?


\[a(\alpha) = g \, \cos(\alpha)\]